400 
PROFESSOR G. H. DARWIN ON FIGURES OF 
Hence it will undoubtedly be more correct to construct tbe surface, of which the 
equation is (72), by regarding the part of r under the symbol 2 as the correction to 
the radius-vector of an ellipsoid of revolution with eccentricity determined by (76), 
where oj 2 /2v is found from (74). 
§ 8. Examples of the Solution. 
The principal object of the preceding investigation is to trace the forms of the two 
masses when they approach to close proximity ; we shall thus be able to determine 
the forms when they are on the point of coalescing into a single mass, and shall 
finally obtain at least an approximate figure of the single mass. For this purpose we 
require to push the approximation by spherical harmonic analysis as far as it will bear. 
We shall below endeavour to estimate the degree of departure from correctness 
involved by the use of this analysis. The results will, therefore, be worked out 
numerically for such values of c/a as bring the two masses close together, and it will 
appear that the largest value of c/a assumed for numerical solution is such that the 
surfaces cross; in this case the reality will be a single mass of a shape which it will 
be possible to draw with tolerable accuracy. 
The computations are facilitated if, instead of assuming c to be an exact multiple of 
a, we take c 2 a multiple of a 2 ; that is to say, we shall take 1/y as an integer, and 
therefore 1//3 also an integer. 
We shall in the first instance suppose the two masses to be equal. In the folio wing- 
examples, then, we have A = a, r = y, B = j8, and the two masses assume the same 
shape. 
The computations will be carried through in detail in two cases, viz., when j8 = y, 
and when = y. The results will also be given for /3 = y. 
When /3 = y, y = }, c/a = 2‘8284, and when /3 = y, y =: y, c/a = 2‘449. Thus 
the distances of the centres apart are 2-f and 2f of the mean radius respectively. 
The numerical details of the two computations may be stated pciri pass'd, and the 
numbers applying to /3 = 1 will be distinguished by being printed in small type. 
In the case of /3 = we have y = y, c/a — 2’6458 ; but only the final result is 
given, without intermediate details. 
The first step is to compute the values of the several series by means of the Tables 
in SS 2 and 5. 
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The numerical results are as follows. 
